Test If A Binary Tree Is Height Balanced ("Balanced Binary Tree" on LeetCode)

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Question: Write a program that takes the root of a binary tree as input and checks whether the tree is height-balanced.

A tree is height balanced if for each node in the tree, the difference in the height of its left and right subtrees is at most one.


Approach 1 (Get Height Of Tree Rooted At Each Node)

We can perform a traversal of the tree and at each node get the height of its left and right subtrees.

This wastes time as we will be repeating work and the traversal of nodes.


Approach 2 (Drill Down With Recursion And Respond Back Up)

We can notice that we don't need to know the heights of all of the subtrees all at once.

All we need to know is whether a subtree is height balanced or not and the height of the tree rooted at that node, not information about any of its descendants.

Our base case is that a null node (we went past the leaves in our recursion) is height balanced and has a height of -1 since it is an empty tree.

So the key is that we will drive towards our base case of the null leaf descendant and deduce and check heights on the way upwards.


Key points of interest:

1.) Is the subtree height balanced?
2.) What is the height of the tree rooted at that node?


Complexities

Time: O( n )

This is a postorder traversal (left right node) with possible early termination if any left subtree turns out unbalanced and an early result bubbles back up.

At worst we will still touch all n nodes if we have no early termination.

Space: O( h )

Our call stack (from recursion) will only go as far deep as the height of the tree, so h (the height of the tree) is our space bound for the amount of call stack frames that we will create

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This question is number 10.1 in "Elements of Programming Interviews" by Adnan Aziz, Tsung-Hsien Lee, and Amit Prakash.